In 1963, weeks before his own assassination, President Kennedy gave approval for the CIA to encourage a group of South Vietnamese generals to depose President Ngo Dinh Diem of South Vietnam. The generals murdered him and his brother on November 1.

Vice President, Lyndon B. Johnson, later said that it was the worst mistake we ever made.

3 Answers
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The quick answer is the Cold War. The context surrounding the assassination of Ngo Dinh Diem provides the long answer.

After WWII the US and the USSR were engaged in the Cold War. The time period from 1949-1963 saw China, North Korea, and Cuba become outright communist countries, and a short lived regime sympathetic to Moscow appeared in Guatemala. This created great fear in the US foreign policy apparatus that the Cold War was being lost one proxy state at a time. This fear is a large contributing factor to why the US had a strong commitment to Ngo Dinh Diem.

In 1963, Ngo Dinh Diem was viewed by many in the US foreign policy apparatus as having lost control of the population, and unable to defeat the communists he was at war with. The famous image of the Buddhist monk lighting himself on fire had just been published in the New York Times, and Kennedy's advisors lost faith in Ngo Dinh Diem.

In this context (fear of the spread of communism) Kennedy gave implicit approval of the assassination operation. According to Stephen Kinzer's book Overthrow Kennedy never gave explicit approval, and regretted the death of Ngo Dinh Diem.

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– CominternSep 24 '14 at 3:36

Question: Why did JFK give approval for the CIA to encourage Diem's "deposal"?

There is no evidence Kennedy ever approved of Diem's assassination. Kennedy gave approval for the American Embassy in Saigon to seek a leadership change. President Diem had survived two previous assassination attempts by his own Military. As a Catholic in a majority Buddhist country who systematically discriminated against Buddhists Diem was kept in power by authoritarian means. These policies in 1963 exploded into executions of hundreds of protesting buddhists across vietnam and this prompted Kennedy to look for alternatives to Diem's 9 year nepotistic authoritarian rule. The mere signalling by Kennedy that the US would accept alternative leadership was enough to incite the Vietnamese military to attempt a third coup d'etat and ultimately assassination. The two highest ranking Vietnamese Generals Dương Văn Minh and Trần Văn Đôn, respectively the Presidential Military Adviser and Army Chief of Staff, led a coup against Diệm. After Diem and his brother(Diem's principal advisor) were arrested while being transported an argument broke out between Diem's brother and Captain Nguyễn Văn Nhung in the back of the transport. After the argument got heated Captain Nguyen bayonetted Diem's brother and then shot Diem in the head killing both men.

The Death of Diem
During the journey back, Nghĩa gave his account of the assassinations to military headquarters: “As we rode back to the Joint General Staff headquarters, Diệm sat silently, but Nhu and the captain [Nhung] began to insult each other. I don’t know who started it. The name-calling grew passionate. The captain had hated Nhu before. Now he was charged with emotion.”[18] When the convoy reached a train crossing, Nghĩa said that Nhung “lunged at Nhu with a bayonet and stabbed him again and again, maybe fifteen or twenty times. Still in a rage, he turned to Diệm, took out his revolver and shot him in the head. Then he looked back at Nhu, who was lying on the floor, twitching. He put a bullet into his head too. Neither Diệm nor Nhu ever defended themselves. Their hands were tied.”

After Diem's death a Military Junta ruled Vietnam for 4 month, before it was overthrown. Captain Nguyen who had killed Diem was himself executed in that coup, he was the only fatality which brought General Nguyễn Khánh to power in 1964.